In such situation, one should do what is the best for them. Not sticking around or running away may be unnatural and thus also emotionally taxing. Giving advice and trying to help, even if the person just wanted to vent, is not bad.

You [1] came to me with your problem. Now it's not only your problem, it's also my problem, since you shared it with me. I want that problem to go away, and will help you or give you advice how to deal with it. Then I will feel better. If I just listen to you, it may be the best thing for you, but for me it is the wrong thing to do, since I'm not made that way.

I you just want to vent so that you can feel better, don't make me feel worse because of that. Find someone else.

[1] "You" and "me" used for clarity, since "they" & "them" is confusing.